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 "Colorings of a right-angled 120-cell.\n",
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11/11/2013 - Permission granted to distribute with attribution.\
\>", "Subsubtitle",
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Cell["The 120-cell", "Section",
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Cell["\<\
The right-angled 120-cell is realized as a union of 14400 copies of a \
hyperbolic 4-simplex.  This 4-simplex is the fundamental domain of the \
hyperbolic Coxeter reflection group with Coxeter diagram 
*-5-*-3-*-3-*-4-*
So we start with this Coxeter reflection group and determine the group \
generated by reflections in the sides of the 120-cell as a subgroup of the \
Coxeter group.  The reflections stabilizing the center of the 120-cell gives \
the spherical Coxeter reflection group of order 14400 given by the Coxeter \
diagram
*-5-*-3-*-3-*
the group of symmetries of a 120-cell.  Since we will only be checking \
combinatorial properties of the 120-cell we take a representation in \
4-dimensional Euclidean space.\
\>", "Text",
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Cell["\<\
Begin with the Gram matrix of the symmetry group.\
\>", "Text",
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Take a corresponding system of unit vectors; the dot products of these gives \
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Cell["\<\
This function gives a matrix for the reflection in the plane through the \
origin normal to a given unit vector.\
\>", "Text",
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So we get matrices for the reflections generating the symmetries of a \
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\>", "Text",
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These generators should satisfy the relations defining the Coxeter \
presentation.\
\>", "Text",
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This function generates a finite matrix group from a given set of generators \
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On the other hand, generating a group from permutations of {1,2,..,n} \
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Generate the orbit of a vector under the action of the group given by matrix \
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The centers of the faces of a Euclidean 120-cell are given as the orbit of \
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We can convert the symmetry group of the 120-cell to a permutation group \
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Cell["\<\
Let\[CloseCurlyQuote]s get the combinatorial structure.  First list the \
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Cell["\<\
Which centers are the closest (largest dot product) to face number 1?\
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Cell["\<\
So 12 adjacent faces.  Which centers are closest to both faces 1 and 2?\
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So 5 faces are adjacent to each of two adjacent faces.  Ridges are pentagons. \
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So the edge in common between faces 1, 2, and 3 ends at the vertices in \
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other edges around the ridge determined by faces 1, 2?\
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So 3 to 4 to 6 to 8 to 5 back to 3 are the faces adjacent to both 1 and 2 in \
cyclic order.  Which sides determine the other 6 ridges of side 1, those \
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and 5?\
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Cell["\<\
So the dodecahedron that is side 1 has adjacent sides 2 through 13 in a \
configuration (roughly, around a sphere)
               10          
             6      4      7
    12          2     3               13
             8      5      9
                11\
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Cell["\<\
List the neighbors of side 1 in a particular order (not 2 to 13 in order, \
unfortunately) to be compatible with the ordering in the paper.\
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that the corresponding manifold covering the right-angled 120-cell is small \
(the manifold being a ",
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 "-fold covering of the right-angled 120-cell).  In particular, we can use \
colors in a 4-dimensional subspace (to get the smallest possible covering \
manifold of this sort).  The corresponding manifold will be orientable iff \
there is a coloring that uses vectors each with an odd number of 1 bits (odd \
vectors).  The four colors at the first vertex are linearly independent and \
every other color is in the 4-dimensional space spanned by these, so we may \
as well take the first four colors as basis vectors of ",
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spanning a 4-dimensional subspace, mapping the four vectors at the first \
vertex to basis vectors takes odd vectors in the span to odd vectors in ",
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Cell["\<\
In a coloring of the 120-cell giving a manifold, the coloring of sides \
adjacent to a fixed dodecahedral side define a coloring of the dodecahedron \
giving a 3-manifold which is a totally geodesic submanifold of the 120-cell \
manifold.  We would also like to find 120-cell manifolds which are relatively \
small and have a small totally geodesic submanifolds from some side.  For \
these we may seed the the search by a defined coloring of a dodecahedron \
assigned to sides adjacent to a first side, taking an appropriate independent \
color for the first side.\
\>", "Text",
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Cell["\<\
We implement an incremental search strategy.  Starting from a partial \
coloring, take an uncolored side and assign it some color from a defined set \
of allowed colors, check that the colors at each vertex of the newly colored \
side are independent and backtrack when a vertex has nonindependent colors, \
or recursively extend the partial coloring, stopping when all sides are \
properly colored.  We extend the coloring of sides in a fixed sequential \
order, so the color of side n must be excluded from the span of colors of \
sides numbered less than n.  We create a table, for each n, of the sets of \
sides with indices less than n that are concurrent at a vertex of side n.  \
The union of the spans of the colorings for these sets constitute the colors \
unavailable for side n.  The fours sides at a given vertex will then be \
linearly independent since the second is not in the span of the first, the \
third not in the span of the first two, and the fourth is not in the span of \
the first three.\
\>", "Text",
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Cell["\<\
Thus there is no check made for the first side color, the second must be \
independent of the first, the third independent of the first two (and of each \
of these separately),  adding the fourth requires checking independence of \
the first three (and each subset of these).  Coloring side 8 completes \
coloring of two vertices, one in common with sides 1, 2, and 5, the other in \
common with 1, 2, and 6.  \
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Cell[TextData[{
 "The search strategy is as follows.  The argument x is a list of rules n->c \
for side n getting color c, coded as a binary integer, for an initial set of \
sides. The list availablecodes gives the colors allowed, a small subset of ",
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manifold.  The variables count and nexti are globally updated and can be \
monitored for the progress of the search by including Dynamic occurrences of \
the variables in the notebook.  The variable result gets the final rule list \
and the search aborted if all sides have been colored.  At each stage, given \
a partial coloring x, for nexti the first side not colored, colorings of each \
set in the constrantsets[[nexti]] are evaluated by substituting x, and the \
colors in each set are Xor\[CloseCurlyQuote]ed together to determine one \
color not available for this side.  The complement of these colors in the set \
of availablecolors gives the valid colorings for this side.  We try each one \
in turn recursively, extending the assignments x with a new color assignment. \
 Of course, if a partial coloring for sides 1 to n leads to an impossible \
coloring for side n+k, then this strategy doesn\[CloseCurlyQuote]t discover \
the failure before trying colors for sides n+1 through n+k-1, and it tries \
all combinations for those sides with failure at side n+k even though \
changing colors of n+1 to n+k-1 doesn\[CloseCurlyQuote]t help the failure at \
n+k.  A smarter strategy would cut down the search tree, but keeping for a \
simple strategy, precomputing the checks to be made, and making those checks \
efficient somewhat offsets the inefficiency of a naive search, especially \
when we hope there is a solution."
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to an orientable 3-manifold.  Is this a totally geodesic 3-manifold in a \
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sides 2-13 (the dodecahedron numbering is defined by ddhfacenumto1nhbrs) and \
color side 1 a fourth basis vector."
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Analogous tables can be produced for the remaining 24 small nonorientable \
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